Such kind of terms also exist in the valence-bond-solid phase as the
redution of continuous symmetry down to discrete symmetry, i.e., \(Z_{3}\)  and
\(Z_{4}\)  symmetry[1]}, [2]}, [3]}. According to Landau criterion, the transition should be
first-order in the presence of cubic terms of order parameter in the
Lagrangian[4]}. In general space-time dimenisions \(D\) , the cubic coupling have
canonical dimensions \([b]=3-D/2\) , which implies that the cubic terms is
strongly relevant near upper critical dimensions \(D^{uc}=4\) . By contrast,
the possible \(Z_{4}\) -anisotropy \(\sim \chi ^{4}+\chi ^{\ast 4}\)  on square
lattice is marginal and can be accessible within \(\epsilon \)  expansion near
four dimensional space-time[5]}. Though the cubic term is relevant at leading
order, in the following, we will show it is irrelevant in the one-loop
corrections. This leaves the concept of FIQCP in the fermion
systems[6]}, [1]}, [2]}.
